CombinatorialCase011 - Revision history

WikiSysop at 21:49, 21 October 2017

2017-10-21T21:49:37Z

← Older revision		Revision as of 17:49, 21 October 2017
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	⍝ M Multicombinations of N items		⍝ M Multicombinations of N items
	⍝ Unlabeled balls, labeled boxes, any # Balls per Box		⍝ Unlabeled balls, labeled boxes, any # Balls per Box
	~~011 1‼3~~ 3		11 2‼3 3
	1 1 1		1 1 1
	1 1 2		1 1 2
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	2 3 3		2 3 3
	3 3 3		3 3 3
	~~011 1‼3~~ 2		11 2‼3 2
	1 1 1		1 1 1
	1 1 2		1 1 2
	1 2 2		1 2 2
	2 2 2		2 2 2
	~~011 1‼3~~ 1		11 2‼3 1
	1 1 1		1 1 1
	</pre>		</pre>

WikiSysop at 21:00, 21 October 2017

2017-10-21T21:00:14Z

← Older revision		Revision as of 17:00, 21 October 2017
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	* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), any # balls per box (1)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), any # balls per box (1)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
			* Allows Lexicographic and Gray Code order
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is an integer matrix.		* Generated result is an integer matrix.
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	<pre>		<pre>
	~~011~~ 1‼2 3		11 1‼2 3 ⍝ Multicombinations in unspecified order
			1 1
			2 2
			1 2
			3 3
			2 3
			1 3
			11 2‼2 3 ⍝ Multicombinations in Lexicographic order
	1 1		1 1
	1 2		1 2
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	2 3		2 3
	3 3		3 3
			11 3‼2 3 ⍝ Multicombinations in Gray Code order
			1 1
			2 2
			1 2
			3 3
			2 3
			1 3
	⍝ M Multicombinations of N items		⍝ M Multicombinations of N items
	⍝ Unlabeled balls, labeled boxes, any # Balls per Box		⍝ Unlabeled balls, labeled boxes, any # Balls per Box

WikiSysop at 22:06, 14 May 2017

2017-05-14T22:06:45Z

← Older revision		Revision as of 18:06, 14 May 2017
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	This case produces '''<apll>L</apll> Multicombinations of <apll>R</apll> items'''. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.		This case produces '''<apll>M</apll> Multicombinations of <apll>N</apll> items'''. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>M</apll>, from the values <apll>⍳N</apll>.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), any # balls per box (1)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is an integer matrix.		* Generated result is an integer matrix.

	The count for this function is <apll>L!L+R-1</apll>.		The count for this function is <apll>M!M+N-1</apll>.

	For example:		For example:
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	2 3		2 3
	3 3		3 3
	⍝ L Multicombinations of R items		⍝ M Multicombinations of N items
	⍝ Unlabeled balls, labeled boxes, any # Balls per Box		⍝ Unlabeled balls, labeled boxes, any # Balls per Box
	011 1‼3 3		011 1‼3 3
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	<pre>		<pre>
	010 ~~1‼L R~~ ↔ (011 ~~1‼L~~,R-L-1)+[⎕IO+1] 0..L-1		010 1‼M N ↔ (011 1‼M,N-M-1)+[⎕IO+1] 0..M-1
	011 ~~1‼L R~~ ↔ (010 ~~1‼L~~,L+R-1)-[⎕IO+1] 0..L-1		011 1‼M N ↔ (010 1‼M,M+N-1)-[⎕IO+1] 0..M-1
	</pre>		</pre>

WikiSysop at 01:45, 30 April 2017

2017-04-30T01:45:14Z

← Older revision		Revision as of 21:45, 29 April 2017
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	This case produces <apll>L</apll> ~~'''~~Multicombinations~~'''~~ of <apll>R</apll> items. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.		This case produces '''<apll>L</apll> Multicombinations of <apll>R</apll> items'''. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)

WikiSysop at 01:17, 30 April 2017

2017-04-30T01:17:12Z

← Older revision		Revision as of 21:17, 29 April 2017
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	3 3		3 3
	⍝ L Multicombinations of R items		⍝ L Multicombinations of R items
	⍝ Unlabeled balls, labeled boxes, any #~~bpb~~		⍝ Unlabeled balls, labeled boxes, any # Balls per Box
	011 1‼3 3		011 1‼3 3
	1 1 1		1 1 1

WikiSysop at 22:39, 29 April 2017

2017-04-29T22:39:47Z

← Older revision		Revision as of 18:39, 29 April 2017
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	This case produces <apll>L</apll> ~~multicombinations~~ of <apll>R</apll> items. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.		This case produces <apll>L</apll> '''Multicombinations''' of <apll>R</apll> items. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)

WikiSysop: Created page with "This case produces L multicombinations of R items. A multicombination is a collection of Multisets (sets which may contain repeated elements) ac..."

2017-04-29T21:36:40Z

Created page with "This case produces <apll>L</apll> multicombinations of <apll>R</apll> items. A multicombination is a collection of Multisets (sets which may contain repeated elements) ac..."

New page

This case produces <apll>L</apll> multicombinations of <apll>R</apll> items. A multicombination is a collection of [[Multisets]] (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length <apll>L</apll>, from the values <apll>⍳R</apll>.

* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), any # balls per box (1)
* Sensitive to <apll>⎕IO</apll>
* Counted result is an integer scalar
* Generated result is an integer matrix.

The count for this function is <apll>L!L+R-1</apll>.

For example:

If we have 2 unlabeled balls (●●) and 3 labeled boxes (<apll>123</apll>) with any # of balls per box, there are <apll>6</apll> (<apll>↔ 2!2+3-1</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|● ●
|   
|   
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|  ●
|  ●
|   
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|  ●
|   
|  ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|   
|● ●
|   
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|   
|  ●
|  ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|   
|   
|● ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|}

The diagram above corresponds to:

<pre>
011 1‼2 3
1 1
1 2
1 3
2 2
2 3
3 3
⍝ L Multicombinations of R items
⍝ Unlabeled balls, labeled boxes, any #bpb
011 1‼3 3
1 1 1
1 1 2
1 1 3
1 2 2
1 2 3
1 3 3
2 2 2
2 2 3
2 3 3
3 3 3
011 1‼3 2
1 1 1
1 1 2
1 2 2
2 2 2
011 1‼3 1
1 1 1
</pre>

In general, this case is related to that of Combinations ([[CombinatorialCase010|<apll>010</apll>]]) via the following identities:

<pre>
010 1‼L R ↔ (011 1‼L,R-L-1)+[⎕IO+1] 0..L-1
011 1‼L R ↔ (010 1‼L,L+R-1)-[⎕IO+1] 0..L-1
</pre>